Suppose we have a rank deficient of $k$ different covariance matrices $R_{k} \in C^{N \times N}$.Different $R_{k}$ are computed by utilizing the Toeplitz structure and they are all positive semidefinite covariance matrices. Also noted that the $R_{1}$ are not orthogonal to $R_{2}$ and not orthogonal to $R_{K}$ and so no, but they are only semi orthogonal.

I want to design a unified (combined) matrix $X$ that can be used to minimize the mean square error (MSE) of the following,

MSE=$\sum_{k=1}^{K}\text{trace}(R_{k}-R_{k}X(X^{H}R_{k}X+{I})^{-1} X^{H}R_{k}$

It has been found that $X$ can be designed based on superposing the eigenvectors of each $R_{k}$ and then summing them up over all $R_{k}$ i.e,

$X=\sum_{k=1}^{K} U_{k}$,

where $U_{k}$ is the eignvectors of $R_{k}$ arranging from the largest ot the lowest eigenvalues.

Since the $R_{k}$ are not completely orthogonal, the obtained $X$ is not completely orthogonal to each of $R_{k}$. Therefore the MSE is increased.

Does anyone have any idea how can I improve the results by select more appropriate $X$?

Best regards.